In an inflationary economy, the purchasing power of your currency continually deteriorates. If you have an investment that grows slower than your purchasing power shrinks, you’re actually losing value over time. The true rate of growth is the rate of change in the purchasing power of your investiment.
If the inflation rate is r and the rate of growth from your investment is i, then intuitively the true rate of growth is
g = i − r.
But is it? That depends on whether your investment and inflation compound periodically or continuously [1].
Periodic compounding
If you start with an investment P, the amount of currency in the investment after compounding n times will be
P(1 + i)n.
But the purchasing power of that amount will be
P(1 + i)n (1 + r)−n.
If the principle were invested at the true rate of growth, its value at the end of n periods would be
P(1 + g)n.
So setting
P(1 + i)n (1 + r)−n = P(1 + g)n
gives us
g = (i − r) / (1 + r).
The true rate of growth is less than what intuition would suggest. To achieve a true rate of growth g, you need i > g, i.e.
i = g + rg + r.
Continuous compounding
With continuous compounding, an invent of P for time T becomes
P exp(iT)
and has purchasing power
P exp(iT) P exp(−rT).
If
P exp(iT) P exp(−rT) = P exp(gT)
then
g = i − r
as expected.
So what?
It’s mathematically interesting that discrete and continuous compounding work differently when inflation is taken into account. But there are practical consequences.
Someone astutely commented that inflation really compounds continuously. It does, and not at a constant rate, either. But suppose we find a value of the monthly inflation rate r equivalent to the true annual rate. And suppose you’re in some sort of contract that pays monthly interest i. Then your true rate of growth is (i − r) / (1 + r), not (i − r).
If r is small, the difference between (i − r) / (1 + r) and (i − r) is small. But the larger r is, the bigger the difference is. As I’ve written about before, hyperinflation is counterintuitive. When r is very large, (i − r) / (1 + r) is much less than (i − r).
Related posts
[1] Robert C. Thompson. The True Growth Rate and the Inflation Balancing Principle. The American Mathematical Monthly, Vol. 90, No. 3 (Mar., 1983), pp. 207–210